论文标题
随机电磁场中的Vlasov-Poisson和Vlasov-Poisson-Fokker-Planck系统：局部适应性良好
The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness
论文作者
论文摘要
在本文中，我们构建了弗拉索夫 - 波森（VP）和弗拉索夫 - 波森 - 福克克（VPFP）系统的独特，本地强的解决方案（VPPFP）系统，该系统受到外部，空间，定期的，白色的时间电气电磁场的$ \ \ Mathbb t^d \ Mathbb r^d $。使用$σ> d/2 + 1 $（除多项式速度权重）时，采用初始条件。我们还表明，如果外力场平稳，则VPFP的解决方案是立即由于低纤维正则化而导致的$ c^\ infty_ {x，v} $。动力学方程式出现了外部强迫，作为速度的随机传输，这意味着与非线性的$ x $和$ v $之间的各向异性相同，即本地理论比可比较的流体机械机械方程更为复杂，该方程比发出了任何添加性随机强迫或随机运输。尽管在血浆物理学文献中经常讨论随机电磁场，但据我们所知，这是对非线性随机动力学方程的强溶液的首次数学研究。
In this paper, we construct unique, local-in-time strong solutions to the Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected to external, spatially regular, white-in-time electromagnetic fields in $\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^σ$ with $σ> d/2 + 1$ (in addition to polynomial velocity weights). We additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due to hypoelliptic regularization if the external force fields are smooth. The external forcing arises in the kinetic equation as a stochastic transport in velocity, which means, together with the anisotropy between $x$ and $v$ in the nonlinearity, that the local theory is a little more complicated than comparable fluid mechanics equations subjected to either additive stochastic forcing or stochastic transport. Although stochastic electromagnetic fields are often discussed in the plasma physics literature, to our knowledge, this is the first mathematical study of strong solutions to nonlinear stochastic kinetic equations.
